Measure (mathematics)

Overview

In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the base set. Depending on the application, the "size" of a subset may be interpreted as (for example) its physical size, the amount of something that lies within the subset, or the probability that some random process will yield a result within the subset. The main use of measures is to define general concepts of integration over domains with more complex structure than intervals of the real line. Such integrals are used extensively in probability theory, and in much of mathematical analysis.

It is often not possible or desirable to assign a size to all subsets of the base set, so a measure does not have to do so. There are certain consistency conditions that govern which combinations of subsets it is allowed for a measure to assign sizes to; these conditions are encapsulated in the auxiliary concept of a σ-algebra.

Measure theory is that branch of real analysis which investigates σ-algebras, measures, measurable functions and integrals.

File:Measure illustration.png

Informally, a measure maps sets to non-negative real numbers, with larger sets being mapped to bigger numbers.

Definition

Formally, a measure μ is a function defined on a σ-algebra Σ over a set X and taking values in the extended interval [0,∞] such that the following properties are satisfied:

• The empty set has measure zero:

<math> \mu(\varnothing) = 0 </math>.

• Countable additivity or σ-additivity: if <math>E_1, E_2, E_3,\,\!</math> ... is a countable sequence of pairwise disjoint sets in <math>\Sigma</math>, the measure of the union of all the <math>E_i\,\!</math>'s is equal to the sum of the measures of each <math>E_i\,\!</math>:

<math> \mu\left(\bigcup_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \mu(E_i).</math>

The triple (X,Σ,μ) is then called a measure space, and the members of Σ are called measurable sets.

A probability measure is a measure with total measure one (i.e., μ(X)=1); a probability space is a measure space with a probability measure.

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki(2004) and a number of other authors. For more details see Radon measure.

Properties

Several further properties can be derived from the definition of a countably additive measure.

Monotonicity

<math>\mu</math> is monotonic: If <math>E_1</math> and <math>E_2</math> are measurable sets with <math>E_1\subseteq E_2</math> then <math>\mu(E_1) \leq \mu(E_2)</math>.

Measures of infinite unions of measurable sets

<math>\mu</math> is subadditive: If <math>E_1</math>, <math>E_2</math>, <math>E_3</math>, ... is a countable sequence of sets in <math>\Sigma</math>, not necessarily disjoint, then

<math>\mu\left( \bigcup_{i=1}^\infty E_i\right) \le \sum_{i=1}^\infty \mu(E_i)</math>.

<math>\mu</math> is continuous from below: If <math>E_1</math>, <math>E_2</math>, <math>E_3</math>, ... are measurable sets and <math>E_n</math> is a subset of <math>E_{n+1}</math> for all n, then the union of the sets <math>E_n</math> is measurable, and

<math> \mu\left(\bigcup_{i=1}^\infty E_i\right) = \lim_{i\to\infty} \mu(E_i)</math>.

Measures of infinite intersections of measurable sets

<math>\mu</math> is continuous from above: If <math>E_1</math>, <math>E_2</math>, <math>E_3</math>, ... are measurable sets and <math>E_{n+1}</math> is a subset of <math>E_n</math> for all n, then the intersection of the sets <math>E_n</math> is measurable; furthermore, if at least one of the <math>E_n</math> has finite measure, then

<math> \mu\left(\bigcap_{i=1}^\infty E_i\right) = \lim_{i\to\infty} \mu(E_i)</math>.

This property is false without the assumption that at least one of the <math>E_n</math> has finite measure. For instance, for each n ∈ N, let

<math> E_n = [n, \infty) \subseteq \mathbb{R} </math>

which all have infinite measure, but the intersection is empty.

Sigma-finite measures

Main article: Sigma-finite measure

A measure space (X,Σ,μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a union of sets with finite measure.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to separability of topological spaces.

Completeness

A measurable set X is called a null set if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

Examples

Some important measures are listed here.

• The counting measure is defined by μ(S) = number of elements in S.
• The Lebesgue measure is the unique complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0,1]) = 1.
• Circular angle measure is invariant under rotation.
• The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property.
• The Hausdorff measure which is a refinement of the Lebesgue measure to some fractal sets.
• Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0,1]). Such a measure is called a probability measure. See probability axioms.
• The Dirac measure <math>\mu_a</math> (confer Dirac delta function) is given by <math>\mu_a(S) = \chi_S(a)</math>, where <math>\chi_S</math> is the characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other measures include: Borel measure, Jordan measure, Ergodic measure, Euler measure, Gauss measure, Baire measure, Radon measure.

Non-measurable sets

Main article: Non-measurable set

Not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.

Generalizations

For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used mainly in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term "positive measure" is used.

Another generalization is the finitely additive measure. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L^∞ and the Stone-Čech compactification. All these are linked in one way or another to the axiom of choice.

The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in <math>\mathbb{R}^n</math> consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k=0,1,2,...,n, and linear combinations of those "measures". "Homogeneous of degree k" means that rescaling any set by any factor <math>c>0</math> multiplies the set's "measure" by <math>c^k</math>. The one that is homogeneous of degree n is the ordinary n-dimensional volume. The one that is homogeneous of degree n-1 is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogeneous of degree 0 is the Euler characteristic.

See also

• Outer measure
• Inner measure
• Hausdorff measure
• Product measure
• Pushforward measure
• Lebesgue measure
• Vector measure
• Almost everywhere
• Lebesgue integration
• Caratheodory extension theorem
• Measurable function
• Steinhaus theorem
• Geometric measure theory

References

• R. G. Bartle, 1995. The Elements of Integration and Lebesgue Measure. Wiley Interscience.
• Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III.
• R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.
• Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0-471-317160-0 Second edition.
• D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
• Paul Halmos, 1950. Measure theory. Van Nostrand and Co.
• R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of Economics, v. 3, pp. 428-32.
• M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.
• Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.
• Some useful Cambridge Tripos Notes on Probability and Measure Theory link